Dependence on parameters of CW globalizations of families of Harish-Chandra modules and the meromorphic continuation of $C^{\infty}$ Eisenstein series

Nolan R. Wallach

Published 2019-02-05Version 1

The first main result is that the Casselman-Wallach Globalization of a real analytic family of Harish-Chandra modules is continuous in the parameter and the family of globalizations is holomorphic if the family of Harish-Chandra modules is holomorphic. Our proof of this result uses results from the thesis of Vincent van der Noort in several critical ways. In his thesis the holomorphic result was proved in the case when the parameter space is a one dimensional complex manifold up to a branched covering. The second main result is a proof of the meromorphic continuation of $C^{\infty}$ Eisenstein series.using Langlands' results in the $K$ finite case as an application of the methods in the proof of the first part.